# Implicit differentiation to find the length of a spiral

• January 9th 2011, 06:30 AM
fatlucky
Implicit differentiation to find the length of a spiral
Hey,
I was wondering if there was a quicker/better way of working this out:

x(t)=(t+a)cos(t), y(t)=(t+a)sin(t)
therefore dx/dt= -(t+a)sin(t) + acos(t)
and dy/dt= (t+a)cos(t) + asin(t)

To work out the length I would have to use L=int{sqrt(1+ [dy/dx]^2 )}
EDIT: forgot about the integral :D

[dy/dx]^2 is a nightmare to work out
do I have to crunch the numbers or is there a way to simplify and do it quicker?

EDIT: Oh.. and this is called parametric differentiation :S
• January 9th 2011, 06:34 AM
alexmahone
$s=\int_{t_1}^{t_2}\sqrt{(\frac{dx}{dt})^2+(\frac{d y}{dt})^2}dt=\int_{t_1}^{t_2}\sqrt{(t+a)^2+a^2}dt$
• January 9th 2011, 06:38 AM
fatlucky
Quote:

Originally Posted by alexmahone
$s=\int\sqrt{(\frac{dx}{dt})^2+(\frac{dy}{dt})^2}dt$

Thanks :D
A useful one to remember